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LESSON 24.5 Properties and Conditions for Kites and Trapezoids Name HARDBOUND SE PAGE 1005 BEGINS HERE Class 24.5 Properties and Conditions for Kites and Trapezoids Essential Question: What are the properties of kites and trapezoids? Resource Locker Exploring Properties of Kites Explore Common Core Math Standards A kite is a quadrilateral with two distinct pairs of congruent consecutive sides. _ _ _ _ _ _ In the figure, PQ ≅ PS, and QR ≅ SR, but QR ≇ QP. The student is expected to: G-SRT.5 Q Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. T P Mathematical Practices Measure the angles made by the sides and diagonals of a kite, noticing any relationships. Language Objective The diagonals of a kite are perpendicular; the diagonals of an isosceles trapezoid are congruent; a kite has exactly one pair of congruent opposite angles; an isosceles trapezoid has two pairs of congruent base angles. © Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©Larry Mulvehill/Corbis Explain to a partner how to describe the properties of kites and trapezoids. Essential Question: What are the properties of kites and trapezoids? A Use a protractor to measure ∠PTQ_ and ∠QTR in the figure. What do your results tell you _ about the kite’s diagonals, PR and QS? _ _ m∠PTQ = 90°, m∠QTR = 90°; the diagonals PR and QS are perpendicular. B Use a protractor to measure ∠PQR and ∠PSR in the figure. How are these opposite angles related? ∠PQR ≅ ∠PSR C Measure ∠QPS and ∠QRS in the figure. What do you notice? ∠QPS ≇ ∠QRS, so only one pair of opposite angles in the kite are congruent. D Use a compass to construct your own kite figure on a separate sheet of paper. Begin by choosing a point B. Then use your compass to choose points A and C so that AB = BC. E Now change the compass length and draw arcs from both points A and C. Label the intersection of the arcs as point D. C B D Mark the intersection of the diagonals as point E. Module 24 E D A A be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction Lesson 5 1241 gh “File info” made throu Date Class ns for d Conditio perties an 24.5 Pro and Trapezoids Kites Name oids? and trapez problems rties of kites es to solve the prope for triangl What are ity criteria Question: and similar Essential congruence figures. G-SRT.5 Use tric of Kites in geome relationships Properties utive sides. Exploring ent consec of congru Explore t pairs _ _ two distinc . _ QP with ≇ QR _ lateral _ a quadri ≅ SR, but _ PS A kite is , and QR PQ ≅ In the figure, Q T HARDCOVER PAGES 10051018 Resource Locker and to prove Turn to these pages to find this lesson in the hardcover student edition. R P S kite, nals of a and diago tell you the sides your results s made by the angle . What do Measure onships. in the figure any relati and ∠QTR _ ndicular. noticing re ∠PTQ_ _ QS are perpe ctor to measu _ QS? Use a protra diagonals, PR and nals PR and the diago kite’s ite TR = 90°; about the these oppos 90°, m∠Q . How are m∠PTQ = the figure ∠PSR in ∠PQR and re ctor to measu Use a protra ? angles related ∠PSR ? ruent. ∠PQR ≅ do you notice are cong What kite . s in the the figure and ∠QRS in site angle the sides ∠QPS and pair of oppo Finally, draw Measure only one ss of the kite. ∠QRS, so e the compa diagonals ∠QPS ≇ Now chang from of the draw arcs intersection uct your length and Label Mark the ss to constr A and C. as point E. Use a compa on a separate sheet both points arcs diagonals figure ction of the a point interse ing own kite the choos e Begin by ss to choos of paper. as point D. your compa C BC. = B. Then use that AB and C so points A D y ⋅ Image g Compan Credits: ©Larry Harcour n Mifflin © Houghto Mulvehill/Corbis t Publishin B C C D E A B B A A Lesson 5 1241 Module 24 4L5.indd 62_U9M2 ESE3897 IN1_MNL Lesson 24.5 Finally, draw the sides and diagonals of the kite. B B A IN1_MNLESE389762_U9M24L5.indd 1241 1241 F C C PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo. Ask students to describe the spider web and in particular its geometrical properties. Then preview the Lesson Performance Task. R S MP.6 Precision ENGAGE Date 1241 4/18/14 8:49 PM 4/18/14 8:51 PM G HARDBOUND SE Measure the angles of the kite ABCD you constructed in Steps D–F and the measure of the angles formed by the diagonals. Are your results the same as for the kite PQRS you used in Steps A–C? PAGE 1006 BEGINS HERE Yes, for kite ABCD, m∠AEB = 90° and m∠CEB = 90°, so, again, the diagonals of the kite EXPLORE Exploring Properties of Kites are perpendicular. Also, for the constructed kite, ∠ABC ≇ ∠ADC, but ∠BAD ≅ ∠BCD, so, again, one pair of opposite angles are congruent but the other pair are not. INTEGRATE TECHNOLOGY Students have the option of doing the kite activity either in the book or online. Reflect 1. In the kite ABCD you constructed in Steps D–F, look at ∠CDE and ∠ADE. What do you notice? _ Is this true for ∠CBE and ∠ABE as well? How can you state this in terms of diagonal AC and the pair of non-congruent opposite angles ∠CBA and ∠CDA? _ ∠CDE ≅ ∠ADE; yes, ∠CBE ≅ ∠ABE; so, diagonal BD bisects the pair of non- QUESTIONING STRATEGIES congruent opposite angles ∠CBA and ∠CDA. 2. What kind of triangles do the diagonals of a kite form? Are any of these triangles congruent? Explain. Right triangles; yes; there are two pairs of congruent triangles by the HL Triangle Congruence Theorem. _ _ In the kite ABCD you constructed _ _ in Steps D–F, look at EC and EA. What do you notice? Is this true for EB and ED as well? Which diagonal is a perpendicular bisector? _ _ _ _ _ EC ≅ EA; no, EB ≇ ED; so, diagonal BD is the perpendicular bisector of _ diagonal AC, but not vice versa. Explain 1 EXPLAIN 1 Using Relationships in Kites The results of the Explore can be stated as theorems. Using Relationships in Kites Four Kite Theorems © Houghton Mifflin Harcourt Publishing Company If a quadrilateral is a kite, then its diagonals are perpendicular. If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. If a quadrilateral is a kite, then one of the diagonals bisects the pair of non-congruent angles. If a quadrilateral is a kite, then exactly one diagonal bisects the other. Q P T R INTEGRATE MATHEMATICAL PRACTICES Focus on Math Communication MP.3 Point out to students that in the previous lesson they were introduced to the properties of rectangles, rhombuses, and squares. Explain that in this lesson, they will be given a quadrilateral and will learn what conditions can be used to classify it as a kite or a trapezoid. You may want to call on students to read each theorem aloud. Then ask them to explain the theorem in their own words. Challenge students to come up with unique ways to explain the theorems. S You can use the properties of kites to find unknown angle measures. Module 24 1242 Lesson 5 PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U9M24L5 1242 9/30/14 1:07 PM Math Background INTEGRATE TECHNOLOGY A kite is a quadrilateral with two distinct pairs of congruent consecutive sides. A trapezoid is a quadrilateral with at least one pair of parallel sides. These definitions may not be the same as definitions in other textbooks. Such decisions about definitions are somewhat arbitrary. Variations of definitions do not change the facts of mathematics, but they do change the way the facts are expressed. The decision to use an inclusive definition for trapezoid means that all parallelograms share the properties of trapezoids. The decision to use an exclusive definition for kite means that other quadrilaterals do not necessarily share the properties of kites. Have students use geometry software to draw the figures in some of the examples. This will allow them to check their answers. Properties and Conditions for Kites and Trapezoids 1242 QUESTIONING STRATEGIES HARDBOUND SE Example 1 PAGE 1007 BEGINS HERE How do you use the properties of a kite to find the measure of its angles? Since the diagonals of a kite are perpendicular, they form right angles. That makes the acute angles of the right triangles complementary. B In kite ABCD, m∠BAE = 32° and m∠BCE = 62°. Find each measure. E A m∠CBE Use angle relationships in △BCE. C D m∠ABE △ABE is also a right triangle. Use the property that the diagonals of a kite are perpendicular, so m∠BEC = 90°. Therefore, its acute angles are complementary. △BCE is a right triangle. m∠ABE + m∠ BAE = 90 ° EXPLAIN 2 m∠BCE + m∠CBE = 90° Substitute 32° for m∠ BAE , then solve for m∠ABE. Proving that Base Angles of Isosceles Trapezoids Are Congruent m∠CBE = 28° Therefore, its acute angles are complementary. Substitute 62° for m∠BCE, then solve for m∠CBE. 62° + m∠CBE = 90° m∠ABE + 32 ° = 90 ° m∠ABE = 58 ° Reflect 3. INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 You may want to review the Parallel Postulate the sum of m∠ABE and m∠CBE. Your Turn 4. Determine m∠ADC in kite ABCD. ∠ADC ≅ ∠ABC , since exactly one pair of opposite angles are congruent. m∠ADC = m∠ABC = m∠ABE + m∠CBE = 58° + 28° = 86° Proving that Base Angles of Isosceles Trapezoids Are Congruent Explain 2 © Houghton Mifflin Harcourt Publishing Company and the Corresponding Angles Theorem before you present a plan for the proof. The Parallel Postulate guarantees that there is a unique line through one vertex of the trapezoid that is parallel to one leg of the trapezoid. The segment determined by this line creates a parallelogram, which in turn creates an isosceles triangle (with congruent base angles). The Corresponding Angles Theorem applies to the figure because the lines are parallel, and the transitive property will give congruent base angles. From Part A and Part B, what strategy could you use to determine m∠ADC? One pair of opposite angles in ABCD is congruent, so m∠ADC = m∠ABC. Also, m∠ABC is A trapezoid is a quadrilateral with at least one pair of parallel sides. The pair of parallel sides of the trapezoid (or either pair of parallel sides if the trapezoid is a parallelogram) are called the bases of the trapezoid. The other two sides are called the legs of the trapezoid. A trapezoid has two pairs of base angles: each pair consists of the two angles adjacent to one of the bases. An isosceles trapezoid is one in which the legs are congruent but not parallel. Isosceles trapezoid Trapezoid base leg leg leg base base base leg Module 24 1243 Lesson 5 COLLABORATIVE LEARNING IN1_MNLESE389762_U9M24L5.indd 1243 Peer-to-Peer Activity Ask students to work with a partner to make a physical model of a kite and of a trapezoid with paper strips and brads. Have one student make a conjecture about how to find the other three angle measures for the kite, and ask the other student to make a conjecture about how to find the other three angle measures for the trapezoid. Ask them to confirm each other’s conjecture by measuring the angles. 1243 Lesson 24.5 6/10/15 9:29 AM HARDBOUND SE Three Isosceles Trapezoid Theorems PAGE 1008 BEGINS HERE If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. How is the isosceles triangle in this proof used to show that the base angles of the isosceles trapezoid are congruent? Sample answer: The congruent sides of the isosceles triangle are used to show that its base angles are congruent. Then the proof establishes that the base angles of the isosceles trapezoid are congruent by the transitive property. If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. A trapezoid is isosceles if and only if its diagonals are congruent. You can use auxiliary segments to prove these theorems. Example 2 Complete the flow proof of the first Isosceles Trapezoid Theorem. Given: an isosceles _ABCD _ is_ _ trapezoid with BC ǁ AD, AB ≅ DC. B C Prove: ∠A ≅ ∠D A E QUESTIONING STRATEGIES D BC || AD Given Draw CE || BA intersecting AD at E. Parallel Postulate ∠A ≅ ∠CED ABCE is a parallelogram. Definition of parallelogram Corresponding Angles Theorem AB ≅ EC Opposite sides of a parallelogram are congruent. AB ≅ DC Given EC ≅ DC Substitution ∠CED ≅ ∠D Isosceles Triangle Theorem ∠A ≅ ∠D Reflect 5. Explain how the auxiliary segment was useful in the proof. Introducing the auxiliary segment breaks the trapezoid into familiar figures, a parallelogram and an isosceles triangle. Then the properties of the simpler figures could be used to prove the theorem about the more complex figure. Module 24 1244 © Houghton Mifflin Harcourt Publishing Company Transitive Property of Congruence Lesson 5 DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U9M24L5 1244 9/30/14 1:10 PM Multiple Representations Have students make a table of the properties of kites and trapezoids. Have them list the properties of each in their own words and draw a diagram to represent each. Then have students compare different types of quadrilaterals. For example, ask: “How are kites and squares alike?” Sample answer: Both are quadrilaterals, and both have perpendicular diagonals. Properties and Conditions for Kites and Trapezoids 1244 6. EXPLAIN 3 to ∠A and ∠C is supplementary to ∠D by the Same–Side Interior Angles Postulate. From Example 2, ∠A ≅ ∠D. So ∠B ≅ ∠C because angles supplementary to ≅ angles are ≅ Using Theorems about Isosceles Trapezoids HARDBOUND SE INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Some math textbooks define a trapezoid as a PAGE 1009 BEGINS HERE Your Turn 7. quadrilateral with exactly one pair of parallel sides. Remind students that this definition is not used here. Parallelograms are a subset of trapezoids because a trapezoid has at least one pair of parallel sides, as they are defined here. Students need to consider this as they are using the theorems about isosceles trapezoids to find segment lengths for trapezoids. If you are trying to find the length of one part of a diagonal of an isosceles trapezoid, what information do you need? You need to know that the diagonals of an isosceles trapezoid are congruent. Can the bases of a trapezoid be congruent? Explain. Yes, if the trapezoid is also a parallelogram. Complete the proof of the second Isosceles Trapezoid Theorem: If a trapezoid has one pair of base angles congruent, then the trapezoid is isosceles. _ _ Given: ABCD is a trapezoid with BC ǁ AD, ∠A ≅ ∠D. B C Prove: ABCD is an isosceles trapezoid. A E _ _ D _ _ _ BA so that CE It is given that BC ǁ AD . By the Parallel Postulate , CE can be drawn parallel to _ intersects AD at E. By the Corresponding Angles Theorem, ∠A ≅ ∠CED . It is given that ∠A ≅ ∠D , _ ¯ ∠CED ≅ ∠D . By the Converse of the Isosceles Triangle Theorem, CE CD . ≅ so by substitution, opposite sides ABCE By definition, is a parallelogram. In a parallelogram, are congruent, so _ _ _ CD _ CE AB ≅ . By the Transitive Property. of Congruence, AB ≅ . Therefore, by definition, ABCD is an isosceles trapezoid . © Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©Mariusz Niedzwiedzki/Shutterstock QUESTIONING STRATEGIES The flow proof in Example 2 only shows that one pair of base angles is congruent. Write a plan for proof for using parallel lines to show that the other pair of base angles (∠B and ∠C) are also congruent. Plan for Proof: In the isosceles trapezoid, the bases are parallel, so ∠B is supplementary Explain 3 Using Theorems about Isosceles Trapezoids You can use properties of isosceles trapezoids to find unknown values. Example 3 Find each measure or value. A railroad bridge has side sections that show isosceles trapezoids. The figure ABCD represents one of these sections. AC = 13.2 m and BE = 8.4 m. Find DE. B C E A D Use the property that the diagonals are congruent. _ _ AC ≅ BD Use the definition of congruent segments. AC = BD Substitute 13.2 for AC. 13.2 = BD Use the Segment Addition Postulate. BE + DE = BD Substitute 8.4 for BE and 13.2 for BD. 8.4 + DE = 13.2 Subtract 8.4 from both sides. DE = 4.8 Module 24 1245 Lesson 5 LANGUAGE SUPPORT IN1_MNLESE389762_U9M24L5 1245 Connect Vocabulary Have students draw a kite and an isosceles trapezoid on poster board and label the diagrams accordingly. For kites, include that their diagonals are perpendicular and that they have one pair of congruent opposite angles. For isosceles trapezoids, include that each pair of base angles is congruent and that the diagonals are congruent. 1245 Lesson 24.5 9/30/14 2:01 PM B Find the value of x so that trapezoid EFGH is isosceles. F HARDBOUND SE (2x2 + 21) PAGE 1010 BEGINS HERE G H E (3x2 - 4) For EFGH to be isosceles, each pair of base angles are congruent. F ∠E ≅ ∠ F are congruent. In particular, the pair at E and m∠E = m∠ F Use the definition of congruent angles. 2 Substitute 3x - 4 for m∠E and 2x + 21 for m∠ F . . 3x 2 - 4 = 2x 2 + 21 2 2 Substract 2x from both sides and add 4 to both sides. Take the square root of both sides. x2 = 25 x= 5 or x = -5 Your Turn 8. In isosceles trapezoid PQRS, use the Same-Side Interior Angles Postulate to find m∠R. 9. Q JL = 3y + 6 and KM = 22 − y. Determine the value of y so that trapezoid JKLM is isosceles. J R P 77° K M L For JKLM to be isosceles, its diagonals must be congruent. _ _ JL ≅ KM S Since PQRS is isosceles, each pair of base angles must be congruent. © Houghton Mifflin Harcourt Publishing Company JL = KM 3y + 6 = 22 - y ∠P ≅ ∠Q; m∠P = m∠Q; 77° = m∠Q 4y = 16 Using the Same-Side Interior Angles Postulate, y=4 m∠Q + m∠R = 180°; 77° + m∠R = 180° m∠R = 103° Explain 4 Using the Trapezoid Midsegment Theorem The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. Module 24 IN1_MNLESE389762_U9M24L5.indd 1246 HARDBOUND SE PAGE 1011 BEGINS HERE midsegment 1246 Lesson 5 4/18/14 8:50 PM Properties and Conditions for Kites and Trapezoids 1246 Trapezoid Midsegment Theorem EXPLAIN 4 The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. _ _ _ _ XY ∥ BC , XY ∥ AD Using the Trapezoid Midsegment Theorem B X D You can use the Trapezoid Midsegment Theorem to find the length of the midsegment or a base of a trapezoid. In the formula for the length of the midsegment, how do you know which segment lengths to substitute where? Sample answer: The midsegment length will be the length inside the trapezoid connecting the midpoints of the legs; the base lengths will be the lengths of the parallel sides. Find each length. Example 4 E In trapezoid EFGH, find XY. X F 12.5 10.3 H Use the second part of the Trapezoid Midsegment Theorem. Substitute 12.5 for EH and 10.3 for FG. Students may have trouble keeping track of given information, especially when algebraic expressions are involved. Encourage students to color code measures on each diagram. This may help students identify the information needed for their calculations. G Y 1 (EH + FG) XY = _ 2 1 (12.5 + 10.3) =_ 2 = 11.4 Simplify. AVOID COMMON ERRORS L In trapezoid JKLM, find JM. Q M 8.3 9.8 K PAGE 1012 BEGINS HERE © Houghton Mifflin Harcourt Publishing Company P HARDBOUND SE Use the second part of the Trapezoid Midsegment Theorem. Substitute 9.8 for PQ and 8.3 for KL . Multiply both sides by 2. J 1 _ KL PQ = ( + JM) 2 1 8.3 + JM 9.8 = _ ( ) 2 19.6 = 8.3 + JM 11.3 = JM 8.3 from both sides. Subtract Your Turn 10. In trapezoid PQRS, PQ = 2RS. Find XY. PQ = 2RS 16.8 = 2RS Q Y 16.8 P R X Module 24 IN1_MNLESE389762_U9M24L5.indd 1247 Lesson 24.5 Y A 1 (BC + AD) XY = _ 2 QUESTIONING STRATEGIES 1247 C S 8.4 = RS 1 XY = (PQ + RS) 2 1 = (16.8 + 8.4) 2 = 12.6 _ _ 1247 Lesson 5 4/18/14 8:50 PM Elaborate ELABORATE 11. Use the information in the graphic organizer to complete the Venn diagram. Rectangle Four right angles Parallelogram Two pairs of parallel sides Rhombus Four congruent sides Quadrilateral Square Four right angles and four congruent sides QUESTIONING STRATEGIES Why can’t theorems about the base angles of an isosceles trapezoid be written as biconditionals while theorems about the diagonals can? Sample answer: One starts with an isosceles trapezoid and results in finding two pairs of base angles congruent; the other starts with a trapezoid and one pair of congruent base angles and reasons the trapezoid is isosceles. So, the hypothesis and conclusion of the two theorems are not the reverse of each other. Trapezoid At least one pair of parallel sides Kite Two distinct pairs of congruent sides Quadrilateral Parallelogram Two pairs of parallel sides Rectangle Four right angles Square Rhombus Four right angles and four congruent sides Four congruent sides SUMMARIZE THE LESSON Kite Two distinct pairs of congruent sides Have students list the properties of kites and trapezoids and the difference between a trapezoid and an isosceles trapezoid. Sample answer: The diagonals of a kite are perpendicular; the diagonals of an isosceles trapezoid are congruent; a kite has exactly one pair of congruent opposite angles; an isosceles trapezoid has two pairs of congruent base angles. An isosceles trapezoid has legs that are congruent but not parallel. Trapezoid At least one pair of parallel sides HARDBOUND SE 12. Discussion The Isosceles Trapezoid Theorem about congruent diagonals is in the form PAGE 1013 of a biconditional statement. Is it possible to state the two isosceles trapezoid theorems BEGINS HERE about base angles as a biconditional statement? Explain. No; the hypotheses and conclusions are not reverses. One has “isosceles trapezoid” as a hypothesis and “each pair of base angles are ≅” as a conclusion; the other has “a trapezoid and one pair of ≅ base angles” as a hypothesis and “isosceles trapezoid” as a conclusion. © Houghton Mifflin Harcourt Publishing Company What can you conclude about all parallelograms? Possible answer: All parallelograms are quadrilaterals and all parallelograms are also trapezoids. 13. Essential Question Check-In Do kites and trapezoids have properties that are related to their diagonals? Explain. Yes; the diagonals of a kite are perpendicular, while the diagonals of an isosceles trapezoid are congruent. Module 24 IN1_MNLESE389762_U9M24L5.indd 1248 1248 Lesson 5 4/18/14 8:50 PM Properties and Conditions for Kites and Trapezoids 1248 Evaluate: Homework and Practice EVALUATE B In kite ABCD, m∠BAE = 28° and m∠BCE = 57°. Find each measure. E A C D ASSIGNMENT GUIDE 1. Concepts and Skills Practice Explore Exploring Properties of Kites Exercises 16–18 Example 1 Using Relationships in Kites Exercises 1–4 Example 2 Proving that Base Angles of Isosceles Trapezoids Are Congruent Exercises 5–6 Example 3 Using Theorems about Isosceles Trapezoids Exercises 7–10 Example 4 Using the Trapezoid Midsegment Theorem Exercises 11–14 4. m∠ADC m∠ABC = m∠ABE + m∠CBE ∠ADC ≅ ∠ABC = 95° m∠ADC = 95° m∠ADC = m∠ABC = 62° + 33° Using the first and second Isosceles Trapezoid Theorems, complete the proofs of each part of the third Isosceles Trapezoid Theorem: A trapezoid is isosceles if and only if its diagonals are congruent. Prove part 1: If a trapezoid is isosceles, then its diagonals are congruent. B Given: ABCD _ _ _ is an _isosceles trapezoid with BC ∥ AD , AB ≅ DC . _ _ Prove: AC ≅ DB A C E F D _ _ , It is given that AB ≅ DC . By the first Trapezoid Theorem, ∠BAD ≅ ∠CDA ― ― and by the Reflexive Property of Congruence, AD ≅ AD . By the SAS Triangle _ _ Congruence Theorem, △ABD ≅ △DCA, and by CPCTC , AC ≅ DB . Exercise IN1_MNLESE389762_U9M24L5 1249 Lesson 24.5 m∠CBE = 33° m∠ABC Module 24 1249 57° + m∠CBE = 90° m∠ABE = 62° 3. m∠CBE m∠BCE + m∠CBE = 90° m∠ABE + m∠BAE = 90° © Houghton Mifflin Harcourt Publishing Company realize how to begin solving the problem. Point out that they must first determine if the figure is a kite, a trapezoid, or an isosceles trapezoid. Then suggest that they redraw each figure, marking known properties for that type of quadrilateral. 2. m∠ABE + 28° = 90° 5. INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 For some exercises, some students may not m∠ABE Lesson 5 1249 Depth of Knowledge (D.O.K.) Mathematical Practices 9/30/14 2:06 PM 1–4 1 Recall of information MP.4 Modeling 5–6 2 Skills/Concepts MP.3 Logic 7–19 2 Skills/Concepts MP.4 Modeling 20–21 2 Skills/Concepts MP.4 Modeling 22–24 2 Skills/Concepts MP.2 Reasoning 25 3 Strategic Thinking MP.3 Logic 26 3 Strategic Thinking MP.3 Logic 6. HARDBOUND SE Prove part 2: If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles. _ _ _ _ Given: ABCD is a trapezoid with BC ∥ AD and diagonals AC ≅ DB . Prove: ABCD is an isosceles trapezoid. B A E F Statements _ _ 2. BE ∥ CF _ _ 3. BC ∥ AD 4. BCFE is a parallelogram. _ _ 5. BE ≅ CF _ _ 6. AC ≅ DB Reasons 1. There is only one line through a given point perpendicular to a given line, so each auxiliary line can be drawn. 2. Two lines perpendicular to the same line are parallel. 3. Given 4. Definition of parallelogram (Steps 2, 3) 5. If a quadrilateral is a parallelogram, then its opposite sides are congruent. 6. Given 7. Definition of perpendicular lines 8. △BED ≅ △CFA 8. HL Triangle Congruence Theorem (Steps 5–7) 9. ∠BDE ≅ ∠CAF 9. CPCTC 10. ∠CBD ≅ ∠BDE, ∠BCA ≅ ∠CAF 10. Alternate Interior Angles Theorem 11. ∠CBD ≅ ∠BCA 11. Transitive Property of Congruence (Steps 9, 10) 12. 12. Given 14. SAS Triangle Congruence Theorem (Steps 12, 13) 15. ∠BAC ≅ ∠CDB 15. CPCTC 17. ABCD is isosceles. Module 24 IN1_MNLESE389762_U9M24L5.indd 1250 © Houghton Mifflin Harcourt Publishing Company 13. Reflexive Property of Congruence 14. △ABC ≅ △DCB 16. ∠BAD ≅ ∠CDA INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Encourage students to develop their own work patterns when they analyze the many types of quadrilaterals in the exercises, especially when algebraic expressions are involved. For example, they may want to color code the measures on each diagram to help them identify the information needed for their calculations. D 7. ∠BED and ∠CFA are right angles. _ _ 13. BC ≅ BC BEGINS HERE C _ _ _ _ 1. Draw BE ⊥ AD and CF ⊥ AD. ― ― AC ≅ DB PAGE 1014 16. Angle Addition Postulate 17. If a trapezoid has one pair of base angles congruent, then the trapezoid is isosceles. 1250 Lesson 5 4/18/14 8:50 PM Properties and Conditions for Kites and Trapezoids 1250 HARDBOUND SE Use the isosceles trapezoid to find each measure or value. PAGE 1015 BEGINS HERE 7. LJ = 19.3 and KN = 8.1. Determine MN. K Find the positive value of x so that trapezoid PQRS is isosceles. 8. L Q (2x2 + 5) J 2x 2 + 5 = x 2 + 30 x 2 = 25 S M ― L J ≅ KM ; L J = KM; 19.3 = KM; KN + MN = KM; 8.1 + MN = 19.3; MN = 11.2 9. m∠Q = m∠P R N ― ∠Q ≅ ∠P In isosceles trapezoid EFGH, use the Same-Side Interior Angles Postulate to determine m∠E. x=5 (x2 + 30) P 10. AC = 3y + 12 and BD = 27 - 2y. Determine the value of y so that trapezoid ABCD is isosceles. _ _ AC ≅ BD A E F H G AC = BD 3y + 12 = 27 - 2y 5y = 15 B 137° y=3 ∠G ≅ ∠H; m∠G = m∠H; 137° = m∠H C m∠E + m∠H = 180°; m∠E + 137° = 180°; m∠E = 43° D Find the unknown segment lengths in each trapezoid. 12. In trapezoid EFGH, find FG. © Houghton Mifflin Harcourt Publishing Company 11. In trapezoid ABCD, find XY. 13.3 D C A XY = X Y 17.9 E B 2 2 13. In trapezoid PQRS, PQ = 4RS. Determine XY. G Q Y 7.4 = 2JK 3.7 = JK Q 18.4 Module 24 IN1_MNLESE389762_U9M24L5.indd 1251 7.4 J S _ _1 (JK + LM) 2 1 7.4 = _(3.7 + LM) PQ = K PQ = 4RS; 18.4 = 4RS; 4.6 = RS 1 1 XY = (PQ + RS) = (18.4 + 4.6) = 11.5 2 2 _ 17.0 = FG PQ = 2JK M X 2 39.4 = 22.4 + FG 14. In trapezoid JKLM, PQ = 2JK. Determine LM. Q P Lesson 24.5 19.7 H _1 (DC + AB) = _1 (13.3 + 17.9) = 15.6 R 1251 F 22.4 _1 (EH + FG) 2 1 19.7 = _(22.4 + FG) PQ = P P L 2 14.8 = 3.7 + LM 11.1 = LM 1251 Lesson 5 3/3/16 10:31 PM 15. Determine whether each of the following describes a kite or a trapezoid. Select the correct answer for each lettered part. A. Has two distinct pairs of congruent consecutive sides kite trapezoid B. Has diagonals that are perpendicular kite trapezoid C. Has at least one pair of parallel sides kite kite trapezoid trapezoid kite trapezoid D. Has exactly one pair of opposite angles that are congruent E. Has two pairs of base angles AVOID COMMON ERRORS HARDBOUND SE 16. Multi-Step Complete the proof of each of the four Kite Theorems. The proof of each of the four theorems relies on the same initial reasoning, so they are presented here in a single two-column proof. B _ _ _ _ Given: ABCD is a kite, with AB ≅ AD and CB ≅ CD . _ _ Prove: (i) AC ⊥ BD ; E A C ABC ≅ ∠ADC; (ii) ∠_ BAD and ∠BCD; (iii) AC _ bisects ∠ _ (iv) AC bisects BD. PAGE 1016 BEGINS HERE D Statements _ _ _ _ 1. AB ≅ AD, CB ≅ CD _ _ 2. AC ≅ AC Reasons 1. Given 2. Reflexive Property of Congruence 3. SSS Triangle Congruence Theorem (Steps 1, 2) 4. ∠BAE ≅ ∠DAE 4. CPCTC _ _ 5. AE ≅ AE 5. Reflexive Property of Congruence 6. △ABE ≅ △ADE 6. SAS Triangle Congruence Theorem (Steps 1, 4, 5) 7. ∠AEB ≅ ∠AED 7. CPCTC 8. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. 9. ∠ABC ≅ ∠ADC 9. CPCTC (Step 3) 10. ∠BAC ≅ ∠DAC and ∠BCA ≅ ∠DCA 10. CPCTC (Step 3) _ 11. AC bisects ∠BAD and ∠BCD. 11. Definition of angle bisector 12. _ BE ≅ _ DE _ _ 13. AC bisects BD. Module 24 IN1_MNLESE389762_U9M24L5 1252 © Houghton Mifflin Harcourt Publishing Company 3. △ABC ≅ △ADC _ _ 8. AC ⊥ BD Some students may have trouble understanding how a trapezoid is isosceles if and only if its diagonals are congruent. Ask them to break up this biconditional statement into two statements to prove. They should see that if the diagonals are congruent, they can use triangle criteria to show congruent triangles and then use corresponding parts of congruent figures to show the trapezoid is isosceles. Conversely, they can start with an isosceles trapezoid to show triangles are congruent and then use corresponding parts of congruent figures to show the diagonals are congruent. 12. CPCTC (Step 6) 13. Definition of segment bisector 1252 Lesson 5 9/30/14 2:09 PM Properties and Conditions for Kites and Trapezoids 1252 17. Given: JKLN is a parallelogram. JKMN is an isosceles trapezoid. K J Prove: △KLM is an isosceles triangle. N _ _ M 1. JKLN is a parallelogram. (Given); 2. KL ≅ NJ (Opposite sides of a parallelogram are congruent.); _ _ 3. JKMN is an isosceles trapezoid. (Given); 4. NJ ≅ MK (Definition of isosceles trapezoid); _ _ 5. KL ≅ KM (Transitive Property of Congruence); 6. △KLM is an isosceles triangle. (Definition of isosceles triangle) L Algebra Find the length of the midsegment of each trapezoid. 18. 19. 12 4x 3y - 7 y+6 6x _ 1 4x = (12 + 6x) 2 8x = 12 + 6x x = 6, so 4x = 4(6) = 24 HARDBOUND SE PAGE 1017 BEGINS HERE y+3 _( ) 1 y+6= (y + 3) + (3y - 7) 2 2(y + 6) = (y + 3) + (3y - 7) 2y + 12 = 4y - 4 8 = y, so y + 6 = 8 + 6 = 14 20. Represent Real-World Problems A set of shelves fits an attic room with one sloping wall. The left edges of the shelves line up vertically, and the right edges line up along the sloping wall. The shortest shelf is 32 in. long, and the longest is 40 in. long. Given that the three shelves are equally spaced vertically, what total length of shelving is needed? The shelves form a trapezoid and the middle shelf forms its midsegment. 1 middle shelf: (32 in. + 40 in.) = 36 in. Total: 32 + 36 + 40 = 108 in. 2 © Houghton Mifflin Harcourt Publishing Company _ C 21. Represent Real-World Problems A common early stage in making an origami model is known as the kite. The figure shows a paper model at this stage unfolded. The folds create four geometric kites. Also, the 16 right triangles adjacent to the corners of the paper are all congruent, as are the 8 right triangles adjacent to the center of the paper. Find the measures of all four angles of the kite labeled ABCD (the point A is the center point of the diagram). Use the facts that ∠B ≅ ∠D and that the interior angle sum of a quadrilateral is 360°. B A D The 8 congruent central triangles are isosceles right triangles; so, m∠A = 2(45)° = 90°. The four angles at each corner of the paper are congruent, 1 so m∠BCA = m∠DCA = (90°) = 22.5° and m∠C = 22.5° + 22.5° = 45°. 4 Since ∠B ≅ ∠D, m∠B = m∠D = x°; then: 90 + x + 45 + x = 360°, 135 + 2x = 360, and x = 112.5. So m∠B = m∠D = 112.5°. _ Module 24 IN1_MNLESE389762_U9M24L5.indd 1253 1253 Lesson 24.5 1253 Lesson 5 4/18/14 8:49 PM 22. Analyze Relationships The window frame is a regular octagon. It is made from eight pieces of wood shaped like congruent isosceles trapezoids. What are m∠A, m∠B, m∠C, and m∠D in trapezoid ABCD? _ _ Extend BA and CD to their intersection E. Since 1 m∠E = (360°) = 45°, and since ∠B ≅ ∠C, 8 m∠B + m∠C + m∠E = 180° _ B C A D 2m∠B + 45° = 180° 2m∠B = 135° m∠B = 67.5° m∠C = 67.5° By the Same-Side Int. ∠s Post., m∠A + m∠B = 180°, so m∠A = 180° - 67.5° = 112.5°. Since ∠A ≅ ∠D, m∠D = 112.5°. and m∠ADE 23. Explain the Error In kite ABCD, m∠BAE = 66° _ _= 59°. Terrence is trying to find m∠ABC. He knows that BD bisects AC, and that therefore △AED ≅ △CED. He reasons that ∠ADE ≅ ∠CDE, so that m∠ADC = 2(59°) = 118°, and that ∠ABC ≅ ∠ADC because they are opposite angles in the kite, so that m∠ABC = 118°. Explain Terrence’s error and describe how to find m∠ABC . Terrence mistakenly reasoned that ∠ABC ≅ ∠ADC; only one pair of opposite angles in a kite are congruent, and they are adjacent to the bisected diagonal, not the bisecting diagonal. To find A m∠ABC: Since ∠BAE and ∠ABE are complementary, m∠ABE can be found by 90° - 66° = 24°. Then, since △AEB ≅ △CEB so that ∠ABE ≅ ∠CBE, m∠ABC is twice m∠ABE, or 2(24°) = 48°. B E C D 24. Complete the table to classify all quadrilateral types by the rotational symmetries and line symmetries they must have. Identify any patterns that you see and explain what these patterns indicate. Angle of Rotational Symmetry © Houghton Mifflin Harcourt Publishing Company Quadrilateral Number of Line Symmetries kite none non-isosceles trapezoid none 0 isosceles trapezoid none 1 parallelogram 180° 0 rectangle 180° 2 rhombus 180° 2 square 90° 4 1 The quadrilaterals with rotational symmetry are parallelograms and special cases of parallelograms; the more restricted cases of quadrilaterals tend to have more line symmetries, up to the square with 4; there are two pairs of quadrilateral types with the same symmetries, kites and isosceles trapezoids, and rectangles and rhombuses. Module 24 IN1_MNLESE389762_U9M24L5 1254 1254 Lesson 5 9/30/14 2:11 PM Properties and Conditions for Kites and Trapezoids 1254 JOURNAL Have students draw an isosceles trapezoid. Have them label the angle measures in terms of x and write a justification for each measure. HARDBOUND SE H.O.T. Focus on Higher Order Thinking PAGE 1018 BEGINS HERE 25. Communicate Mathematical Ideas Describe the properties that rhombuses and kites have in common, and the properties that are different. Rhombuses and kites both have pairs of congruent consecutive sides, but in a rhombus, this is because all four sides are congruent. In a kite, two distinct pairs of consecutive sides are congruent. The diagonals of both types of quadrilaterals are perpendicular. In a kite, exactly one diagonal is bisected by the other, while in a rhombus, each diagonal bisects the other. Finally, in a kite, exactly one pair of opposite angles are congruent, while both pairs of opposite angles of a rhombus are congruent. B 26. Analyze Relationships In kite ABCD, triangles B B' ABD and CBD can be rotated and translated, _ _ identifying AD with CD and joining the remaining pair of vertices, as shown in the figure. Why is this process guaranteed to produce an isosceles trapezoid? Suggest a process guaranteed to produce a kite from an isosceles trapezoid, using figures to illustrate your A process. C © Houghton Mifflin Harcourt Publishing Company A = D' Module 24 IN1_MNLESE389762_U9M24L5.indd 1255 1255 Lesson 24.5 D = C' Sample answer for the case of producing an D isosceles trapezoid from a kite where the B congruent angles are obtuse: By the kite_and _ the _ _ of a_ _definition Reflexive of Congruence, BA ≅ BC, AD ≅ CD, BA ≇ AD, _ Property _ and BD ≅ BD _ and translate the triangles BAD and BCD so _. Rotate coincide, to produce the quadrilateral BB’DA that sides AD and _ CD _ _ and B'F_ perpendicular to the line containing AD at shown. Draw BE_ points E and F. BE and B'F are parallel because they are perpendicular to the same line. △BAD ≅ △B'C'D' by SSS, so ∠BAD ≅ ∠B'C'D' because corresponding parts of congruent figures are congruent. Then ∠BAE ≅ ∠B'C'F because supplements of congruent angles E A = D’ D = C’ are congruent. _ _∠BEA ≅ ∠B'FC' because they are both right angles. _ _ Also, BA ≅ B'C' (from the kite), so △BAE ≅ △B'C'F by AAS. So, BE ≅ B'F because corresponding parts of congruent figures are congruent. Quadrilateral BB'FE is a parallelogram, because _one pair of _and congruent. By the definition of a parallelogram, BB' _parallel _ opposite sides are and EF (which includes AD (or C'D')) are parallel, so BB'DA is a trapezoid because it has a pair of ‹ › − parallel sides. Consider the lines BA and B'C' cut by transversal EF . Then ∠BAE and ∠B'C'D' are corresponding angles. Because ∠BAD is obtuse, its supplement, ∠BAE, is_ acute. But ∠B'C'D' is_ obtuse, which means that ∠BAE and ∠B'C'D' cannot be congruent. Thus, BA is not parallel to B'C' and BB'DA is not a parallelogram. A trapezoid that is not a parallelogram but has congruent legs is isosceles, so BB'DA is an isosceles trapezoid. L = K' Sample an isosceles trapezoid, K L _ answer: _ JKLM _ is_ _ with JK ≅ ML but JK ≇ KL or JM. Rotate and _ translate _ triangles JKL and MLK so that sides JL and M'K' coincide. This can be done because the M diagonals of an isosceles trapezoid are congruent. J K The resulting figure has two distinct pairs of _ _ _ _ congruent sides, JK ≅ M'L' (given) and KL ≅ L'K' (Reflexive Property of Congruence), so the figure is a kite. 1255 B’ F L' J = M' Lesson 5 6/10/15 9:36 AM Lesson Performance Task INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 In the model of the spider web, CD = 6 cm This model of a spider web is made using only isosceles triangles and isosceles trapezoids. and BE = 3 cm. Without referring to the circumference of either the outer or the inner circle, explain how you know that the radius of the inner circle is half the radius of the outer circle. By the converse of the Triangle Midsegment Theorem, B is the midpoint of ¯ AC. So, ¯ AB, a radius of the inner circle, has half the measure of ¯ AC, a radius of the outer circle. A B E C D a. All of the figures surrounding the center of the web are congruent to figure ABCDE. Find m∠A. Explain how you found your answer. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections _ MP.1 Explain how you know that BE is parallel _ b. Find m∠ABE and m∠AEB. c. Find m∠CBE and m∠DEB. d. Find m∠C and m∠D. a. 22.5°; 16 triangles congruent to △ABE surround the center of the web, making up a total of 360°. 360° ÷ 16 = 22.5° to CD. Sample answer: m∠ABE = 78.75° = m∠C, so ∠ABE ≅ ∠C. So, ¯ BE ǁ ¯ CD because corresponding angles ∠ABE and ∠C are congruent. © Houghton Mifflin Harcourt Publishing Company b. m∠A = 22.5° and △ABE is an isosceles triangle. So, ∠ABE ≅ ∠AEB and m∠ABE = m∠AEB 180° = m∠ABE + m∠AEB + 22.5° 180° = m∠ABE + m∠ABE + 22.5° 157.5° = 2 ⋅ m∠ABE 78.75° = m∠ABE Thus, each angle measures 78.75°. c. Because the angles form a linear pair, then m∠ABE + m∠CBE = 180°. So 78.75° + m∠CBE = 180° and m∠CBE = 101.25°. In an isosceles trapezoid, base angles are congruent, so m∠DEB = 101.25°. ― ― d. In isosceles trapezoid CBED, BE ‖ CD. Then corresponding angles are congruent and ∠ABE ≅ ∠C and ∠AEB ≅ ∠D. So m∠C = m∠D = 78.75°. Module 24 1256 Lesson 5 EXTENSION ACTIVITY IN1_MNLESE389762_U9M24L5 1256 The Lesson Performance Task deals with triangles and trapezoids in spider webs. Have students research other examples that illustrate quadrilaterals in the animal world, the plant world, or both. For each example students find, they should make a sketch, identify the quadrilateral, and provide additional information they feel is relevant, particularly as it relates to geometry. 10/17/14 11:31 PM Scoring Rubric 2 Points: Student correctly solves the problem and explains his/her reasoning. 1 Point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 Points: Student does not demonstrate understanding of the problem. Properties and Conditions for Kites and Trapezoids 1256